Of GREATEST ATTRACTION. 223 



The value of <p, now found, is remarkable for being a near 

 approximation to any arch of which t is the tangent, provided 

 that arch do not exceed 45 °. The lefs the arch is, the more 

 near is the approximation ; but the exprefiion can only be con- 

 fidered as accurate when <p — o. 



This will be made evident by comparing the fraction 



^" """" — —L with the feries, that gives the arch in terms of the 

 9+ S* 



t % t J t 7 



tangent t, viz. <p zz t -f- — K &c. The fraction 



3 5 7 



-^-7 — -jr- — * -f — r -f, &c. Ihe two firft 



9 + 5' 3 2 7 3-9" 



terms of thefe feries agree ; and in the third terms, the differ- 

 ence is inconfiderable, while t is lefs than unity ; but the agree- 

 ment is never entire, unlefs t zz o, when both feries vanifli. 



The attraction, therefore, or the gravitation at the pole of 

 an oblate fpheroid, is not a maximum, until the eccentricity of 

 the generating ellipfis vanifh, and the fpheroid pafs into a 

 fphere. 



From the circumftance of the value of <p above found, agree- 

 ing nearly with an indefinite number of arches, we muit con- 

 clude, that when a fphere paffes into an oblate fpheroid, its at- 

 traction varies at firft exceeding flowly, and continues to do fo 

 till its oblatenefs, or the eccentricity of the generating ellipfis, 

 become very great. This may be fhewn, by taking the value 

 of F, and fubftituting in it that of <p, in terms of tan ?. 



We have FzzAUL . tan^—p. and fince ^ _ tan ^ _ 



™r~4 tan? 3 

 col <p 3 



tan $ 



